Singular plane sections and the conics in the Fermat quintic threefold

TL;DR

This paper derives explicit equations for conics in the Fermat quintic threefold, linking plane sections with singular points to bitangent lines, and demonstrates their use in studying conic families.

Contribution

It provides the first explicit equations for conics in the Fermat quintic threefold and explores their relation to singular plane sections and bitangent lines.

Findings

Explicit equations for conics in the Fermat quintic threefold.

A birational correspondence between plane sections and bitangent lines.

Identification of special one-dimensional conic families.

Abstract

We present explicit equations for the space of conics in the Fermat quintic threefold $X$, working within the space of plane sections of $X$ with two singular marked points. This space of two-pointed singular plane sections has a birational morphism to the space of bitangent lines to the Fermat quintic threefold, which in its turn is birational to a 625-to-1 cover of $\PP^4.$ We illustrate the use of the resulting equations in identifying special cases of one-dimensional families of conics in $X.$